few_shots: Example:
Problem/activity:
Narrative
The purpose of this activity is for students to create composite shapes and compose new shapes from composite shapes. For example, students may find that a rectangular prism can be made from four triangular prisms or from eight smaller cubes. Students then discover that they can combine two of the composed rectangular prisms to make a new rectangular prism.
Required Materials
Materials to Gather
Geoblocks
Launch
Groups of 4
Give each group geoblocks.
Display a large cube.
“This cube can be built from other shapes. See how many ways you can build a cube using other shapes.”
3 minutes: partner work time
Share responses.
Activity
“Choose a different shape. Try to find smaller shapes you can use to build that shape.”
5 minutes: partner work time
Student Response
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Activity Synthesis
Display a cube and a composite cube—for example, a cube composed of four small cubes and one rectangular prism.
Demonstrate putting the shapes together.
“What new shape did I make with the cube and the cube I made?”
“Put your shapes together to make a rectangular prism.”
Display a rectangular prism and two triangular prisms.
“Find these shapes. Put them together. What new shape can you make?” (a cube, a rectangular block)


Standard options: 
A. Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.
B. Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.
C. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. (Students do not need to learn formal names such as “right rectangular prism.”)


Thought: This activity asks students to compose three-dimensional shapes, using geoblocks, to create composite shapes.
Answer: C

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_lesson_009431

1
options: Use square root and cube root symbols to represent solutions to equations of the form x^(2) = p and x^(3) = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that Ö2 is irrational.

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.

Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^(2) × 3^(–5) = 3^(–3) = 1/3^(3) = 1/27.

Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
problem_activity: Activity
In previous activities and lessons, students found the exact area of a square in order to find an approximation for the square root of an integer. In this activity, students start with a square root of an integer, and draw a square to verify that a given approximation of the square root is reasonable. This is the first time students have seen or drawn squares that do not have vertices at the intersection of grid lines, so it may take them a few minutes to make sense of the new orientation.
Teacher Notes for IM 6–8 Math Accelerated
After arranging students in groups of 2, display for all to see
\(\sqrt{7}\)
and
\(\sqrt{50}\)
and ask, “What two whole numbers does each square root lie between? Be prepared to explain your reasoning.” After a brief quiet think time, invite students to share their reasoning. For example, students may make a list of perfect squares and then find which two the number was between. Once 2–3 strategies are shared, have students continue with the activity as indicated in the launch.
Launch
Give students access to four-function calculators. Display the diagram for all to see. Ask students what is the same and what is different about this diagram and diagrams they have seen in earlier activities. Arrange students in groups of 2. Give students 2–3 minutes of quiet work time followed by partner and whole-class discussions.
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer for data collection and organizing information.
Supports accessibility for: Language; Organization
Writing, Speaking, Listening: MLR1 Stronger and Clearer Each Time.
After students have had time to identify a point on the number line that is closer to
\(\sqrt{3}\)
, ask them to write a brief explanation of their reasoning. Give students time to meet with 2–3 partners, to share and get feedback on their writing. Display prompts that students can ask each other that will help students strengthen their ideas and clarify their language. For example, “How do you know that
\(\sqrt{3}\)
is between 1.5 and 2?” and “How did you determine the area of the square that you drew?” Students can borrow ideas and language from each partner to refine and clarify their original explanation. This will help students refine both their ideas and their verbal and written output.
Design Principles(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
Diego said that he thinks that
\(\sqrt{3}\approx 2.5\)
.
[IMG]
Use the square to explain why 2.5 is not a very good approximation for
\(\sqrt{3}\)
. Find a point on the number line that is closer to
\(\sqrt{3}\)
. Draw a new square on the axes and use it to explain how you know the point you plotted is a good approximation for
\(\sqrt{3}\)
.
Use the fact that
\(\sqrt{3}\)
is a solution to the equation
\(x^2 = 3\)
to find a decimal approximation of
\(\sqrt{3}\)
whose square is between 2.9 and 3.1.
Student Response
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Student Facing
Are you ready for more?
A farmer has a grassy patch of land enclosed by a fence in the shape of a square with a side length of 4 meters. To make it a suitable home for some animals, the farmer would like to carve out a smaller square to be filled with water, as in the figure.
What should the side length of the smaller square be so that half of the area is grass and half is water?
[IMG]
Student Response
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for free access to Extension Student Response.
Activity Synthesis
Invite one or two students to share their squares. Then tell students, “The square of a point on the number line can be visualized as the area of a literal square. This can help us estimate the value of a square root. Simply squaring the number can as well. Let’s check the squares of some numbers that are potential approximations of
\(\sqrt{3}\)
.”
Then ask students to suggest decimal approximations, and check together as a class by finding their squares. Students should be using each guess to make a better guess next. For example, if they try 1.5, then the square is 2.25, which is too low. This suggests trying bigger. Because we know that 2 is too big (because
\(2^2 = 4\)
, it should be somewhere in between 1.5 and 2.) For example, students might suggest this order:
\(1^2 = 1\)
and
\(2^2=4\)
\(1.5^2 = 2.25\)
\(1.8^2 = 3.24\)
\(1.7^2 = 2.89\)
\(1.72^2 = 2.9584\)
\(1.73^2 = 2.9929\)
So 1.73 is a pretty good approximation of
\(\sqrt{3}\)
.


correct_option_index: 0